Help with non-linear system of equations (Trigonometric and Polynomial)

To: mathgroup at smc.vnet.net

Subject: [mg115578] Help with non-linear system of equations (Trigonometric and Polynomial)

From: Manjinder Benning <manjbenning at gmail.com>

Date: Sat, 15 Jan 2011 04:45:29 -0500 (EST)

Hi i am trying to do some 3D modeling by finding a 6 unknown 6 equation
simultaneous system. I am using the equation:
Cos(a) = A.B/|A||B|
where A and B are lines (vectors) in 3 space.
a is the angle between the lines.
When i run the calculation, Mathematica just runs forever and doesn't come
to a solution. It would be great if a closed form solution could be found.
I am understanding that it is difficult to solve systems with polynomial and
trig parts?
Description of Problem:
So i have 3 points in 3-space.
Pl (Xl, Yl, Zl) - A constant point in the air
Pc(Xc, Yc, Zc) - A constant point on the ground (Z=0)
Pe's (Xe_n, Ye_n, Ze_n) - A series of points that lay on the edges of an
ellipse co-planar to Z=0 (which are known experimentally)
A - The angle between the line (Pl -> Pc) and each (Pl -> Pe_n)
Due to the geometry of the problem (a ellipse) angle A is constant.
Are there any methods i can use to make it easier to solve (ie bounding the
problem) or am i dreaming that i will find a closed form solution?
...any help would be appreciated:
Heres the code:
eq1 =Cos[A] == ((Xc - Xl)*(Xe1-Xl) + (Yc-Yl)*(Ye1-Yl) + (-Zl)*(-Zl))/Sqrt[(
(Xc - Xl)^2 + (Yc-Yl)^2+(-Zl)^2 ) * ((Xe1-Xl)^2 +(Ye1-Yl)^2+(-Zl)^2)]
eq2 =Cos[A] == ((Xc - Xl)*(Xe2-Xl) + (Yc-Yl)*(Ye2-Yl) + (-Zl)*(-Zl))/Sqrt[(
(Xc - Xl)^2 + (Yc-Yl)^2+(-Zl)^2 ) * ((Xe2-Xl)^2 +(Ye2-Yl)^2+(-Zl)^2)]
eq3 =Cos[A] == ((Xc - Xl)*(Xe3-Xl) + (Yc-Yl)*(Ye3-Yl) + (-Zl)*(-Zl))/Sqrt[(
(Xc - Xl)^2 + (Yc-Yl)^2+(-Zl)^2 ) * ((Xe3-Xl)^2 +(Ye3-Yl)^2+(-Zl)^2)]
eq4 =Cos[A] == ((Xc - Xl)*(Xe4-Xl) + (Yc-Yl)*(Ye4-Yl) + (-Zl)*(-Zl))/Sqrt[(
(Xc - Xl)^2 + (Yc-Yl)^2+(-Zl)^2 ) * ((Xe4-Xl)^2 +(Ye4-Yl)^2+(-Zl)^2)]
eq5 =Cos[A] == ((Xc - Xl)*(Xe5-Xl) + (Yc-Yl)*(Ye5-Yl) + (-Zl)*(-Zl))/Sqrt[(
(Xc - Xl)^2 + (Yc-Yl)^2+(-Zl)^2 ) * ((Xe5-Xl)^2 +(Ye5-Yl)^2+(-Zl)^2)]
eq6 =Cos[A] == ((Xc - Xl)*(Xe6-Xl) + (Yc-Yl)*(Ye6-Yl) + (-Zl)*(-Zl))/Sqrt[(
(Xc - Xl)^2 + (Yc-Yl)^2+(-Zl)^2 ) * ((Xe6-Xl)^2 +(Ye6-Yl)^2+(-Zl)^2)]
Solve[{eq1,eq2,eq3,eq4,eq5,
eq6},{A,Xl,Yl,Zl,Xc,Yc}]
much gratitude to all things Mathematica!
Manjinder.
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www.thepowerofyang.com